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this article is a part of the Walter Rudin scholar sequence in complicated arithmetic.

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Additional info for Real and Complex Analysis (3rd Edition) (International Series in Pure and Applied Mathematics)

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25 and Theorem 1 . 19(d) to the last inte­ gral in (4) The result is (5) c < 1, we have ex > Is dJJ. for every simple measurable s satisfying 0 < s < f, so that ex > If dJJ.. Since (5) holds for every (6) (7) The theorem follows from ( 1 ), (2), and (7). : X� [0, oo] is measurable, for n f(x) L fn(x) E X), = = 00 1, 2, 3, Ill/ ... , and (x n=l (1) then (2) If djJ. � I djJ.. PROOF First, there are sequences {sa ' { s�'} of simple measurable functions such that s� � f1 and s�' � /2 , as in Theorem 1.

Let C be the complement of Theorem 2. o each p C there corresponds an open set W, such that K W, and p ¢ WP . 7 Theorem c U, U c c V c U. U U. E c G with empty intersection. 6 there are points Pb . . , Pn that E C such The set V = G n WPl n · · · n WPn then has the required properties, since WP l • • • WPn V Ill/ Let f be a real (or extended-real) function on a topological space. If {x:f(x) > ex} is open for every real ex, f is said to be lower semicontinuous. If {x:f(x) < ex} is open for every real f is said to be upper semicontinuous.

I ==1, 2, 3, , f ) i X C (x) ) ( (x) c X v x X. (1) A(V) = lim Jxf dA = lim Jxf dp. ( V) . (2) Now let E be a Borel set in X, and choose 0. 17, there is a closed set F and an open set V such that F E V and . (E) - Jl(E) I for every 0. (E) = JL(E). /Ill In Exercise 18 a compact Hausdorff space is described in which the com­ plement of a certain point fails to be a-compact and in which the conclusion of n - oo 9n 9n n - oo E < E. l > E F) < c E, c � + E. E +E' the preceding theorem is not true.